Superadditivity of Quantum Channel Coding Rate with Finite Blocklength Joint Measurements

The maximum rate at which classical information can be reliably transmitted per use of a quantum channel strictly increases in general with $N$, the number of channel outputs that are detected jointly by the quantum joint-detection receiver (JDR). This phenomenon is known as superadditivity of the maximum achievable information rate over a quantum channel. We study this phenomenon for a pure-state classical-quantum (cq) channel and provide a lower bound on $C_N\mathrm{/}N$, the maximum information rate when the JDR is restricted to making joint measurements over no more than $N$ quantum channel outputs, while allowing arbitrary classical error correction. We also show the appearance of a superadditivity phenomenon—of mathematical resemblance to the aforesaid problem—in the channel capacity of a classical discrete memoryless channel (DMC) when a concatenated coding scheme is employed, and the inner decoder is forced to make hard decisions on $N$-length inner codewords. Using this correspondence, we develop a unifying framework for the above two notions of superadditivity, and show that for our lower bound to $C_N\mathrm{/}N$ to be equal to a given fraction of the asymptotic capacity $C$ of the respective channel, $N$ must be proportional to $V\mathrm{/}{C}^2$, where $V$ is the respective channel dispersion quantity.